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And base on this added the line
>
> <dir>/usr/share/ghostscript</dir>
>
> between <fontconfig> and </fontconfig> of the /etc/fonts/local.conf file.

On my 13.2 system, the fonts are in /usr/share/ghostscript/fonts/, /usr/share/fonts/ghostscript is a symlink to that.
So try to add one of those dirs to /etc/fonts/local.conf or ~/.config/fontconfig/fonts.conf instead.

Btw, on my 13.2 system here the URW fonts show up just fine in KDE, Inkscape, and LibreOffice without any manual change (I don't have a 13.1 system near me to try). So maybe you should revert all your custom modifications to /etc/fonts/?
Try to remove/rename /etc/fonts/ and reinstall the package fontconfig to get back the defaults.
And make sure that ~/.config/fontconfig/fonts.conf doesn't interfere, so better remove/rename this as well.

You might also have to recreate the fontconfig cache: (as root and as user to be sure)

Code:

fc-cache -r
sudo fc-cache -r

PS: "fc-list" shows a list of all fonts found by fontconfig. On my system:

Re: URW fonts disappeared again

PS: I just tried with an unmodified 13.1 Live USB-stick, as I had one lying around.
KDE and LibreOffice showed the URW fonts there as well (couldn't test Inkscape as that isn't included in the KDE Live system).

So this definitely works with the default font configuration even in 13.1...